Properties

Label 21.6.a.c
Level 21
Weight 6
Character orbit 21.a
Self dual yes
Analytic conductor 3.368
Analytic rank 0
Dimension 1
CM no
Inner twists 1

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Newspace parameters

Level: \( N \) \(=\) \( 21 = 3 \cdot 7 \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 21.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(3.36806021607\)
Analytic rank: \(0\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

\(f(q)\) \(=\) \( q + 5q^{2} + 9q^{3} - 7q^{4} + 94q^{5} + 45q^{6} - 49q^{7} - 195q^{8} + 81q^{9} + O(q^{10}) \) \( q + 5q^{2} + 9q^{3} - 7q^{4} + 94q^{5} + 45q^{6} - 49q^{7} - 195q^{8} + 81q^{9} + 470q^{10} + 52q^{11} - 63q^{12} - 770q^{13} - 245q^{14} + 846q^{15} - 751q^{16} - 2022q^{17} + 405q^{18} + 1732q^{19} - 658q^{20} - 441q^{21} + 260q^{22} - 576q^{23} - 1755q^{24} + 5711q^{25} - 3850q^{26} + 729q^{27} + 343q^{28} + 5518q^{29} + 4230q^{30} + 6336q^{31} + 2485q^{32} + 468q^{33} - 10110q^{34} - 4606q^{35} - 567q^{36} - 7338q^{37} + 8660q^{38} - 6930q^{39} - 18330q^{40} - 3262q^{41} - 2205q^{42} + 5420q^{43} - 364q^{44} + 7614q^{45} - 2880q^{46} + 864q^{47} - 6759q^{48} + 2401q^{49} + 28555q^{50} - 18198q^{51} + 5390q^{52} + 4182q^{53} + 3645q^{54} + 4888q^{55} + 9555q^{56} + 15588q^{57} + 27590q^{58} - 11220q^{59} - 5922q^{60} - 45602q^{61} + 31680q^{62} - 3969q^{63} + 36457q^{64} - 72380q^{65} + 2340q^{66} + 1396q^{67} + 14154q^{68} - 5184q^{69} - 23030q^{70} + 18720q^{71} - 15795q^{72} + 46362q^{73} - 36690q^{74} + 51399q^{75} - 12124q^{76} - 2548q^{77} - 34650q^{78} + 97424q^{79} - 70594q^{80} + 6561q^{81} - 16310q^{82} - 81228q^{83} + 3087q^{84} - 190068q^{85} + 27100q^{86} + 49662q^{87} - 10140q^{88} - 3182q^{89} + 38070q^{90} + 37730q^{91} + 4032q^{92} + 57024q^{93} + 4320q^{94} + 162808q^{95} + 22365q^{96} + 4914q^{97} + 12005q^{98} + 4212q^{99} + O(q^{100}) \)

Embeddings

For each embedding \(\iota_m\) of the coefficient field, the values \(\iota_m(a_n)\) are shown below.

For more information on an embedded modular form you can click on its label.

Label \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1.1
0
5.00000 9.00000 −7.00000 94.0000 45.0000 −49.0000 −195.000 81.0000 470.000
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(3\) \(-1\)
\(7\) \(1\)

Inner twists

This newform does not admit any (nontrivial) inner twists.

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 21.6.a.c 1
3.b odd 2 1 63.6.a.b 1
4.b odd 2 1 336.6.a.i 1
5.b even 2 1 525.6.a.b 1
5.c odd 4 2 525.6.d.c 2
7.b odd 2 1 147.6.a.f 1
7.c even 3 2 147.6.e.c 2
7.d odd 6 2 147.6.e.d 2
12.b even 2 1 1008.6.a.a 1
21.c even 2 1 441.6.a.c 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
21.6.a.c 1 1.a even 1 1 trivial
63.6.a.b 1 3.b odd 2 1
147.6.a.f 1 7.b odd 2 1
147.6.e.c 2 7.c even 3 2
147.6.e.d 2 7.d odd 6 2
336.6.a.i 1 4.b odd 2 1
441.6.a.c 1 21.c even 2 1
525.6.a.b 1 5.b even 2 1
525.6.d.c 2 5.c odd 4 2
1008.6.a.a 1 12.b even 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2} - 5 \) acting on \(S_{6}^{\mathrm{new}}(\Gamma_0(21))\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 1 - 5 T + 32 T^{2} \)
$3$ \( 1 - 9 T \)
$5$ \( 1 - 94 T + 3125 T^{2} \)
$7$ \( 1 + 49 T \)
$11$ \( 1 - 52 T + 161051 T^{2} \)
$13$ \( 1 + 770 T + 371293 T^{2} \)
$17$ \( 1 + 2022 T + 1419857 T^{2} \)
$19$ \( 1 - 1732 T + 2476099 T^{2} \)
$23$ \( 1 + 576 T + 6436343 T^{2} \)
$29$ \( 1 - 5518 T + 20511149 T^{2} \)
$31$ \( 1 - 6336 T + 28629151 T^{2} \)
$37$ \( 1 + 7338 T + 69343957 T^{2} \)
$41$ \( 1 + 3262 T + 115856201 T^{2} \)
$43$ \( 1 - 5420 T + 147008443 T^{2} \)
$47$ \( 1 - 864 T + 229345007 T^{2} \)
$53$ \( 1 - 4182 T + 418195493 T^{2} \)
$59$ \( 1 + 11220 T + 714924299 T^{2} \)
$61$ \( 1 + 45602 T + 844596301 T^{2} \)
$67$ \( 1 - 1396 T + 1350125107 T^{2} \)
$71$ \( 1 - 18720 T + 1804229351 T^{2} \)
$73$ \( 1 - 46362 T + 2073071593 T^{2} \)
$79$ \( 1 - 97424 T + 3077056399 T^{2} \)
$83$ \( 1 + 81228 T + 3939040643 T^{2} \)
$89$ \( 1 + 3182 T + 5584059449 T^{2} \)
$97$ \( 1 - 4914 T + 8587340257 T^{2} \)
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